Chessboard and Domino problem

  
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Paul Valle    (2011-01-09 00:42:23)
Chessboard and Domino problem

You are given 31 domino pieces (that each cover two squares of the board) to cover the remaining 62 squares in the ChessPosition (see diagram) below. No matter how you twist and turn those dominoes you can't complete the task. Why?


8
7
6
5
4
3
2
1
a b c d e f g h




Thibault de Vassal    (2011-01-09 00:55:38)
Chessboard and Domino problem

Hoho, a great problem. I started with mathematical reasoning while the solution is sooooo easy :)


Lim Jit Loong    (2011-01-09 13:05:19)
Chessboard and Domino problem

The 2 kings is not next to each other.... so u can see that from this problem file a and h, rank 1 and rank 8 have 7 squares left, so this is impossible to fill up all the squares by domino. In order to fill all the squares with all 31 dominoes, the number of all files and ranks must be even in numbers. This is just what I see from this problem. If any mistakes please correct me.


Thibault de Vassal    (2011-01-09 14:25:31)
Chessboard and Domino problem

Hi Lim, I know someone who tried to make the same reasoning/observation (if I understand yours well), but if the Black King is in b7 (6 lines & 6 columns from the White King), it is still impossible to place the 31 dominos...


Philip Roe    (2011-01-09 18:56:43)
Chessboard and Domino problem

This is actually very easy once you see it. Wherever you place a domino, it covers one black square and one white. So any number of dominoes covers equal numbers of black and white squares. If the two kings are placed on squares of the same color, the task is impossible.

A variation, that I have not yet tried to solve. If the Kings are placed on any two squares of opposite color, can the dominoes always be placed?


Paul Valle    (2011-01-10 02:15:21)
Chessboard and Domino problem

That's right - you always end up with the last domino needing to cover two black squares. I didn't get it at first, until I pictured a miniature version with 4 squares and one domino piece. There should be many solution for each possible variant of two opposite color squares being removed from the chessboard









 

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